Theorem nfcnv 5866

Description: Bound-variable hypothesis builder for converse relation. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypothesis

Ref	Expression
nfcnv.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfcnv	⊢ Ⅎ𝑥◡𝐴

Proof of Theorem nfcnv

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-cnv 5671	. 2 ⊢ ◡𝐴 = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴𝑦}
2		nfcv 2927	. . . 4 ⊢ Ⅎ𝑥𝑧
3		nfcnv.1	. . . 4 ⊢ Ⅎ𝑥𝐴
4		nfcv 2927	. . . 4 ⊢ Ⅎ𝑥𝑦
5	2, 3, 4	nfbr 5160	. . 3 ⊢ Ⅎ𝑥 𝑧𝐴𝑦
6	5	nfopab 5182	. 2 ⊢ Ⅎ𝑥{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴𝑦}
7	1, 6	nfcxfr 2925	1 ⊢ Ⅎ𝑥◡𝐴

This proof depends on syntax axioms:  Ⅎwnfc 2912   class class class wbr 5111  {copab 5175  ^◡ccnv 5662

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-cnv 5671

This theorem is used by:  nfrn  5944  nfpred  6311  nffun  6563  nff1  6776  funcnvmpt  6995  nfsup  9418  nfinf  9450  gsumcom2  20089  ptbasfi  23789  mbfposr  25862  itg1climres  25924  nfwsuc  36345  aomclem8  43846  rfcnpre1  45797  rfcnpre2  45809  smfpimcc  47580